We study problems for which the iterative method \gmr
for solving linear systems of equations makes no progress
in its initial iterations. Our tool for analysis is
a nonlinear system of equations, the stagnation system,
that characterizes this behavior.
For problems of dimension 2 we can solve this system
explicitly, determining that every choice of eigenvalues
leads to a stagnating problem for eigenvector matrices
that are sufficiently poorly conditioned.
We partially extend this result to higher dimensions
for a class of eigenvector matrices called extreme.
We give necessary and sufficient conditions for stagnation
of systems involving unitary matrices, and show
that if a normal matrix stagnates then so does an entire
family of nonnormal matrices with the same eigenvalues.
Finally, we show that there are real matrices for which
stagnation occurs for certain complex right-hand sides
but not for real ones.
(Also UMIACS-TR-2001-74)